1. Fixed-Point Arithmetics: Part II

2. Fixed-Point Notation
A K-bit fixed-point number can be interpreted as either:
an integer (i.e., 20645)
a fractional number (i.e., 0.75)

3. Integer Fixed-Point Representation
N-bit fixed point, 2’s complement integer representation
X = -bN-1 2N-1 + bN-2 2N-2 + … + b020
Difficult to use due to possible overflow
In a 16-bit processor, the dynamic range is
-32,768 to 32,767.
Example:
200 × 350 = 70000, which is an overflow!

4. Fractional Fixed-Point Representation
Also called Q-format
Fractional representation suitable for DSPs algorithms.
Fractional number range is between 1 and -1
Multiplying a fraction by a fraction always results in a fraction and will not produce an overflow (e.g., 0.99 x less than 1)
Successive additions may cause overflow
Represent numbers between
-1.0 and 1 − 2−(N-1), when N is number of bits

5. Fractional Fixed-Point Representation
Also called Q-format
Fractional representation suitable for DSPs algorithms.
Fractional number range is between 1 and -1
Multiplying a fraction by a fraction always results in a fraction and will not produce an overflow (e.g., 0.99 x less than 1)
Successive additions may cause overflow
Represent numbers between
-1.0 and 1 − 2−(N-1), when N is number of bits

6. Fractional Fixed-Point Representation
Equivalent to scaling
Q represents the “Quantity of fractional bits”
Number following the Q indicates the number of bits that are used for the fraction.
Q15 used in 16-bit DSP chip, resolution of the fraction will be 2^–15 or e–6
Q15 means scaling by 1/215
Q15 means shifting to the right by 15
Example: how to represent in memory:
Method 1 (Truncation): INT[0.2625*215]= INT[8601.6] = 8601 =
Method 2 (Rounding): INT[0.2625* ]= INT[8602.1] = 8602 =

7. Truncating or Rounding?
Which one is better?
Truncation
Magnitude of truncated number always less than or equal to the original value
Consistent downward bias
Rounding
Magnitude of rounded number could be smaller or greater than the original value
Error tends to be minimized (positive and negative biases)
Popular technique: rounding to the nearest integer
Example:
INT[251.2] = 251 (Truncate or floor)
ROUND [ 251.2] = 252 (Round or ceil)
ROUNDNEAREST [251.2] = 251

8. Q format Multiplication
Product of two Q15 numbers is Q30.
So we must remember that the 32-bit product has two bits in front of the binary point.
Since NxN multiplication yields 2N-1 result
Addition MSB sign extension bit
Typically, only the most significant 15 bits (plus the sign bit) are stored back into memory, so the write operation requires a left shift by one.
16-bit memory
Q15
Q15
Sign bit X
Extension sign bit
15 bits
15 bits

9. General Fixed-Point Representation
Qm.n notation
m bits for integer portion
n bits for fractional portion
Total number of bits N = m + n + 1, for signed numbers
Example: 16-bit number (N=16) and Q2.13 format
2 bits for integer portion
13 bits for fractional portion
1 signed bit (MSB)
Special cases:
16-bit integer number (N=16) => Q15.0 format
16-bit fractional number (N = 16) => Q0.15 format; also known as Q.15 or Q15

10. General Fixed-Point Representation
N-bit number in Qm.n format:
Value of N-bit number in Qm.n format:
Fixed Point

11. Some Fractional Examples (16 bits)
Integer (15 bits) S
Fraction (15 bits) . .
Q15.0
Q.15 or Q15
Used in DSP
Binary pt position .
Upper 2 bits
Remaining 14 bits
Q1.14

12. How to Compute Fractional Number
Q m.n Format
b’sb’m-1…b’0
bn-1bn-2…b0 .
-2mb’s+…+21b’1+20b’0+2-1bn bn-2…+2-nb0
Examples:
1110 Integer Representation Q3.0: = -2
11.10 Fractional Q1.2 Representation: = = -0.5
(Scaling by 1/22)
1.110 Fractional Q3 Representation: = = (Scaling by 1/23)

13. General Fixed-Point Representation
Min and Max Decimal Values of Integer and Fractional 4-Bit Numbers (Kuo & Gan)

14. General Fixed-Point Representation
Dynamic Range
Ratio between the largest number and the smallest (positive) number
It can be expressed in dB (decibels) as follows:
Dynamic Range (dB) =
Note: Dynamic Range depends only on N
N-bit Integer (Q(N-1).0):
Min = 1; Max = 2N => Max/Min = 2N-1 - 1
N-bit fractional number (Q(N-1)):
Min = 2-(N-1); Max = 1-2-(N-1) => Max/Min = 2N-1 – 1
General N-bit fixed-point number (Qm.n)
=> Max/Min = 2N-1 – 1

15. General Fixed-Point Representation
Dynamic Range and Precision of Integer and Fractional 16-Bit Numbers (Kuo & Gan)

16. General Fixed-Point Representation
Precision
Smallest step (difference) between two consecutive N-bit numbers.
Example:
Q15.0 (integer) format => precision = 1
Q15 format => precision = 2-15
Tradeoff between dynamic range and precision
Example: N = 16 bits
Q15.0 => widest dynamic range (-32,768 to
32,767); worst precision (1)
Q15 => narrowest dynamic range (-1 to +1-); best
precision (2-15)

17. General Fixed-Point Representation
Dynamic Range and Precision of 16-Bit Numbers for Different Q Formats (Kuo & Gan)

18. General Fixed-Point Representation
Scaling Factor and Dynamic Range of 16-Bit Numbers (Kuo & Gan)

19. General Fixed-Point Representation
Fixed-point DSPs use 2’s complement fixed-point numbers in different Q formats
Assembler only recognizes integer values
Need to know how to convert fixed-point number from a Q format to an integer value that can be stored in memory and that can be recognized by the assembler.
Programmer must keep track of the position of the binary point when manipulating fixed-point numbers in asembly programs.

20. How to convert fractional number into integer
Conversion from fractional to integer value:
Step 1: normalize the decimal fractional number to the range determined by the desired Q format
Step 2: Multiply the normalized fractional number by 2n
Step 3: Round the product to the nearest integer
Step 4: Write the decimal integer value in binary using N bits.
Example:
Convert the value 3.5 into an integer value that can be recognized by a DSP assembler using the Q15 format => 1) Normalize: 3.5/4 = 0.875;
2) Scale: 0.875*215= 28,672; 3) Round: 28,672

21. How to convert integer into fractional number
Numbers and arithmetic results are stored in the DSP processor in integer form.
Need to interpret as a fractional value depending on Q format
Conversion of integer into a fractional number for Qm.n format:
Divide integer by scaling factor of Qm.n => divide by 2n
Example:
Which Q15 value does the integer number 2 represent? 2/215=2*2-15=2-14

22. Finite-Wordlength Effects
Wordlength effects occur when wordlength of memory (or register) is less than the precision needed to store the actual values.
Wordlength effects introduce noise and non-ideal system responses
Examples:
Quantization noise due to limited precision of Analog-to-Digital (A/D) converter, also called codec
Limited precision in representing input, filter coefficients, output and other parameters.
Overflow or underflow due to limited dynamic range
Roundoff/truncation errors due to rounding/truncation of double-precision data to single-precision data for storage in a register or memory.
Rounding results in an unbiased error; truncation results in a biased error => rounding more used in practice.

23. Real Floating-Point Numbers
Numbers with fractions
Could be done in pure binary
= =9.625
Where is the binary point?
Fixed?
Very limited
Moving?
How do you show where it is?

24. Floating Point +/- .significand x 2exponent
Biased
Exponent
Sign bit
Significand or Mantissa
+/- .significand x 2exponent
Point is actually fixed between sign bit and body of mantissa
Exponent indicates place value (point position) – used to offset the location of the binary point left or right

25. Floating Point Number Representation
Mantissa is stored in 2’s complement
Exponent is in excess or biased notation
Excess (bias): 127 (single precision); 1023 (double precision) to obtain positive or negative offsets
Exponent field: 8 bits (single precision); 11 bits (double precision) – determines dynamic range
Mantissa: 23 bits (single precision); 52 bits (double precision) – determines precision

26. Floating-Point Number Representation
Floating-point numbers are usually normalized; i.e., exponent is adjusted so that leading bit (MSB) of mantissa is 1
Since MSB of mantissa is always 1, there is no need to store it

27. IEEE 754 Standard for floating point storage 32 and 64 bit standards
8 and 11 bit exponent respectively
Extended formats (both mantissa and exponent) for intermediate results

28. IEEE 754 Formats

29. Floating-point Arithmetic +/-
Check for zeros
Align significands (adjusting exponents)
Add or subtract significands
Normalize result

30. Floating-Point Arithmetic x/
Check for zero
Add/subtract exponents
Multiply/divide significands (watch sign)
Normalize
Round
All intermediate results should be in double length storage